# Quantum G2 at roots of unity: diagrams vs. algebra

For generic $q$, Kuperberg's $G_2$ spider agrees with the category of representations of the quantum group.  What happens when $q$ is a root of unity?  Except for a few small roots of unity, it turns out that the spider agrees with the category of tilting modules for $G_2$, and as a consequence the semisimplified spider agrees with the semisimple quantum group fusion category.  As a corollary of this result together with the trivalent categories classification, we can prove a Kazhdan-Wenzl-style recognition theorem for $G_2$.  This is joint work with Victor Ostrik.